Bifurcation and monotonicity in competition reaction-diffusion systems

Nonlinear

Anolysa.

Theory,

Methods

Pergamon

& Applicnbons, Vol. 23, No. I, pp. l-13. 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/94 $7.00+ .OO

0362-546X(93)E0026-D

BIFURCATION AND MONOTONICITY IN COMPETITION REACTION-DIFFUSION SYSTEMS YIHONG Dut and K. J. BROWNS t Department of Mathematics, University of New England, Armidale, NSW 2351, Australia; and $ Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, U.K. (Received 5 September Key words and phrases:

1991; received for publication

Reaction-diffusion

17 March 1994)

equations, competition models, sub and supersolutions,

global bifurcation theory.

1. INTRODUCTION CONSIDER the system

of reaction-diffusion

equations

u,(x, t) - d, Au(x, t) = au - bu2 - cuu

xED,t>O

u,(x, t) - d2 Av(x, t) = ev - fv2 - guv

where D is a bounded region in R” with smooth boundary, d, , d2, a, b, c, e, f, g are positive numbers. The above system of equations models the situation where two competing species co-exist in D. u(x, t), v(x, t) represent the population densities of the species at time I and at a point x E D; d, , d, give the rates at which the species diffuse; a and e give the birth rates of the species; b and f are self-limiting coefficients; c and g determine the nature of the interaction between the species. For simplicity we shall concern ourselves with the case where u and v satisfy zero Dirichlet boundary conditions on aD but our results could similarly be established for other linear boundary conditions. Clearly only nonnegative solutions are of physical interest. It is natural to seek stationary nonnegative solutions of the above system, i.e. nonnegative solutions of the steady-state problem -d, Au = au - bu2 - cuv,

XED

-d2 Au = ev - fv2 - guv,

XED

u=v=o, The form of this problem the following simpler form

can be simplified.

XEI~D. By a resealing

of u and v, it can be changed

-Au

= au - u2 - cuv,

XED

-Au

= ev - v2 - guv,

XED

u=v=o,

into

(1.1)

x E ao.

The main point of interest about problem (1 .l) is the existence solutions, i.e. (u, v) with u(x) > 0 and u(x) > 0 in D. This problem in [l-6] where many further references can be found.

and multiplicity of positive has been studied extensively

2

Y. Du and K. J. BROWN

In [4], using degree theory, Dancer gives a quite satisfactory description of the positive solution set of (1.1) in the (c, g) plane, where interesting multiplicity results are presented. In [l] by using a decoupling procedure and global bifurcation methods Blat and Brown prove the existence of a continuum of positive solutions in the e - (u, v) plane joining branches of semitrivial solutions (i.e. nonnegative solutions of (1.1) with either u or u identically zero). In this paper we will give a much fuller description of the set of all positive solutions of (1.1) in the e - (u, u) plane. By using the decoupling procedure of [ 11, system (1.1) is changed into an equivalent single equation with a nonlocal nonlinear term. In Section 2, we observe that in the case of competing species the single equation thus obtained has monotonicity properties which permit the use of sub- and supersolution techniques on the equation and, thus, new various results can be obtained about the existence of solutions of (1.1). In Section 3 we combine these results with the results which can be obtained from global bifurcation theory to obtain more detailed information about the set of positive solutions of (1 .l). Although global bifurcation theory is a powerful technique, its results are usually expressed primarily in terms of the existence of continua of solutions and lack precision as, in general, continua may assume much more complicated forms than the simple smooth curves usually seen in bifurcation diagrams. For example, the existence of a continuum of solutions joining points (Ai, u,) and (AZ, u2) in the (A, v) plane guarantees the existence of a solution for I, 5 a I A1 but the existence of a continuum of solutions passing through (A,, v,), (AZ, u2) and (A,, wi) where Ai < & may or may not correspond to the existence of at least two solutions for A1 < 1 < A,. Because of the monotonicity property of the equation, however, we are able to prove more detailed results about the continuum in this case; for example maximal and minimal solutions arising from sub- and supersolution arguments must lie on the continuum and various multiplicity results which are geometrically obvious if the continuum is assumed to be a smooth curve do in fact hold. At the end of this section we show that results from [2,4] on the direction of bifurcation from branches of semitrivial solutions are sufficient to guarantee that various hypotheses used in the statement of theorems in the section are satisfied for appropriate choices of the parameters a, c and g. It is interesting to note that for the corresponding predator-prey system the single equation with nonlocal nonlinearity obtained from the decoupling procedure lacks monotonicity and analogues of our results cannot be obtained. In this case, however, no multiplicity results are known; uniqueness results have been established for various parameter values and it has been conjectured that there exists at most one nontrivial solution for any values of the parameters. 2.

DECOUPLING

AND

MONOTONICITY

We begin by collecting some known results about single equations. LetE = C@(D),fi E (0, l), F = Cd@) = (u E C’(D): ulaD = 0). We shall consider the spaces E and F as being ordered by the usual cones of nonnegative functions PE and PF. Clearly u E int PF if u > 0 in D and au/an < 0 on ao, where n is the outward normal; we shall write u %- 0 in this case. Suppose q E E. Then the linear eigenvalue problem -Au

+ qu = Au u=o

in D on i3D

(2.1)

Reaction-diffusion

systems

has an infinite sequence of eigenvalues, which are bounded below. We denote the lowest eigenvalue of (2.1) by A,(q). It is known that Al(q) is a simple eigenvalue and that the corresponding eigenfunction 6 does not change sign on D. Furthermore, we have

Consider

A,(%) < &(qz)

if q1 < q2,

~1hl)

if qn -+ q in C(D).

now the nonlinear

+ ~,(d

eigenvalue -Au

problem

+ qu = Au - u2

u=o We have the following

in D on dD.

(24,

lemma.

LEMMA 2.1. Problem (2.2), has a positive solution u (i.e. u E PF\(0)) if and only if A > A,(q). Moreover, for each A > A,(q), (2.2), has a unique positive solution ux, the map J. - ux from (Al(q), +co) to F is strongly increasing (i.e. u x, s uxz if A1 > J.,) and continuous.

Proof. Only the conclusions about the map A w ux need some explanation, the others are well known (see, for example, [l]). The fact that A c ux is strongly increasing follows from a suband supersolution argument together with the strong maximum principle. The continuity of A ++ ux follows from the uniqueness of uh and the compactness of the solution set in R’ x C’(B). If q = 0 and A > A,(O), we denote the solution Consider now the system of equations

ux of (2.2), by 13’.

-Au

= au - u2 - cuv,

xeD

-Au

= Au - v2 - guv,

XED

u=v=o,

(2.3),

x E aD

where we have replaced e in (1.1) by 1 to stress that this birth rate is being treated as a parameter whereas a, c and g are taken as fixed positive constants. We shall assume throughout that a > A,(O) so that (2.3), possesses the semitrivial solution (u, v) = (0”, 0) for all A. We now describe the decoupling technique of [ 11. Rewrite -Au

= au - u, - cuv

u=o

in D on aD

as -Au

+ (cu)u = au - u2

on aD.

u=o It follows

from lemma

in D

2.1 that (2.4) has a unique positive solution solution when a I A,(cv). Denote

a > A,(cv), and no positive

u(v) =

U,

if a > A,(cv)

0

if a 5 AI(

(2.4) u, for each v E E when

4

Y. Du and K. J. BROWN

Clearly

(u(u), u) will be a solution -Av

of (2.3), if u satisfies

the single equation in D

= Au - v2 - gu(u)u v=o

Wh

on aD.

Conversely, if (u, v) is any positive solution of (2.3),, then we must have u = u(u) and u satisfies (2.9,. Therefore, the study of the system (2.3), is equivalent to the study of the single equation (2.5),. Clearly, the map u c-) u(v) plays an important role in (2.9,. The following lemma is proved in [ 11. LEMMA

2.2. (i) v c u(v) is continuous from C’(@ (ii) u(u,) 5 2.4~~) if 21i 2 v,; (iii) u(O) = B”, 0 5 u(u) < 8” for any u > 0.

to C’(D);

Define n(v) = g(P - u(v)). Clearly v ++ n(u) is continuous C’(D) and n(0) = 0, 0 < n(v) I g/Y for u > 0. Now consider the following three equations + g0”v = Au - v2 in D,

-Au

and increasing

v = 0 on f3D

-AU

+ gPv

= Lv - v2 + n(v)v in D,

-Au

+ gPv

= ,iv -

v2 + gl?“v in D,

from

C’(O)

to

(2.61,

u = 0 on dD

(2.7),

2, = 0 on aD.

t2.a

Clearly (2.7), is equivalent to (2.5), and is sandwiched between equations (2.6), and (2.8),. It follows from lemma 2.1 that (2.6), has a unique positive solution 0, if and only if ,l > Ii( Equation (2.8), is equivalent to (2.2), with q = 0 and so its unique positive solution is ox, A > A,(O). Moreover, ,J - Ox and A y 8’ are both continuous and strongly increasing. For simplicity we denote A0 = L,(gP) and A, = L,(O) and denote the branches of solutions by L and U (see Fig. 1). It is easy to see that 4: 1 y /z,(c&‘) is an increasing function of /I with r$(/1i) = L,(O) < a. It is proved in [l] that lim,,,~,(c@‘) = to and so there exists a unique value ,J which we denote by A0 such that a = ,l,(c@‘).

t

Y

I

u = ((h. e”))

Fig. 1.

Reaction-diffusion systems LEMMA 2.3. (i) If A < I,,

(ii) Suppose

equation (2.7), has no positive u is any positive solution of (2.7), . Then

solution.

the appropriate inequality holding whenever Ox or 0’ exists. (iii) If A 2 A’, then v = 0’ is a solution of (2.7),. (iv) There exists A, > 0 such that if J. > A0 then u = 8’ is the only positive (2.7), . (v) Suppose

that u # 0” is a solution

of (2.7),.

Then

solution

of

u $4 8”.

Proof. (i) Multiplying (2.7), by u where u is any positive solution and integrating over D proves the result. (ii) Suppose that u is any positive solution of (2.7), and ,l > I,, i.e. Ox exists. Then v is a strict supersolution for (2.6), . It is possible to find an arbitrarily small subsolution ~5of (2.6), and so it follows from the uniqueness of 0, that 4 < Ox < u. A similar argument using the fact that arbitrarily large constants are supersolutions of (2.8), shows that u I 8’ for A > A,. (iii) If 1 2 ,l”, then a = I,(c13~~) 5 ~,(c&‘) and so u(c@) = 0. Thus, u = 0’ is a solution of (2.7), . (iv) Let 4 > 0 denote -A4

the solution

of

+ ge”$ = A04 in D,

C#I= 0 on i3D,

max 4(x) = 1. D

large positive Then for ,l > Ao, (A - A,)$ is a subsolution of (2.6),. Since any sufficiently constant is a supersolution, the unique positive solution Oxof (2.6), must satisfy t$, I (A - A,)$. Then u 2 Bx 2 (A - A,)$. Since Suppose that u is any positive solution of (2.7),. n,(c(J. - A,)$) + CO as A -+ co, there exists ho > 0 such that Ai(c(1 - Ao)4) > a whenever I > A,. Thus, if 13 > A,, A,(cu) 2 Ai(c(A - A,)$) > a and so u(u) = 0, i.e. v = 8’. (v) If J. I A’, the result follows from (ii). Suppose now that A > ho. Since u # Bx, we must have that u(u) f 0 and so a > A,(cu). But a = ~l(cBxo) and the result follows. The solution u = 0’ of (2.7), corresponds to the semitrivial solution (0, tYX)of (2.3),. We shall call a positive solution u # 0’ of (2.7), a nontrivial solution of (2.7), . Clearly if u is a nontrivial solution of (2.7),, then u(u) + 0. Equation (2.7), seems less straightforward than equations (2.6), and (2.8), as it contains a nonlocal term n(u). Since, however, u y n(u) is an increasing function many of the standard proofs of theorems on sub- and supersolutions (see, e.g. Sattinger [7]) still hold for (2.7),. In particular we have the following lemma. LEMMA 2.4. Suppose

that vi 5 u2. Then ur I _v 5 v I v2.

that vi and u2 are sub- and supersolutions, there exist minimal and maximal solutions

Finally in this section we prove the existence certain ranges of A. Let

of maximal

A = (A E R: (2.7), has a positive

respectively, of (2.7), such g and V of (2.7), such that

and minimal

nontrivial

solution

solutions

u).

of (2.7), for

Y. Du and K. J.BROWN

6

Clearly Let

,? E A if and only if (2.3), has a positive A, = infA

and

solution. ?l* = sup A.

It follows from lemma 2.3 that A, 5 A, I A* i A,. It is easy to see that u = 0 is a solution of (2.7), for all A and standard local bifurcation analysis shows that a branch of positive solutions bifurcates from this trivial zero branch if and only if A = &. The direction of bifurcation depends on the parameters a, c and g (we shall discuss this in more detail in Section 3) but it follows that A is nonempty with A* I A,, 5 A*. LEMMA

2.5. Suppose

u, of (2.7),*. of (2.7),, .

Similarly,

that 1, < mini&,, A’). Then there exists a positive nontrivial solution if A* > maxlAo, I.‘], then there exists a positive nontrivial solution U*

Proof. Suppose that there exists no such solution v, . Then there exist sequences A,, + A, + 0 and v, such that v, is a positive solution of (2.7),“. A simple compactness argument shows that there exists a subsequence of u, which converges to some U, in C,‘(o) and that u, L 0 satisfies (2.7),*. Since A, < Ao, we must have that v, # 0 as otherwise A, would be a bifurcation point from the branch of zero solutions of (2.7),. Since A, < Lo, we have a > A,(&‘*) and so u, # ox*. Now suppose A* > maxlAo, A’) and there is no such solution. Then we can find a sequence of positive nontrivial solutions u, of (2.7),, with A,, 4 A* - 0. Again simple compactness arguments show the existence of a positive solution u* of (2.7),, where u* is the limit in Cd@) of a subsequence of u, ; we suppose for simplicity that v, + v*. Since u, L ox, then we must have that u* # 0. If u* = I!?‘*, then A,(cv,) + Ai(c@*) > a and so u(v,) = 0 for n large, a contradiction. This completes the proof. 2.6. (i) Suppose A, < A’. Then for every A E (A,, A’), equation (2.7), has a maximal positive solution v’. If, in addition, A* < Ao, then (2.7),* also has a maximal positive solution. (ii) Suppose A* > I,. Then for every A E (Ao, A*) equation (2.7), has a minimal positive solution vx with ux # 8’. If, in addition, A* > A’, then (2.7),, also has a minimal positive solution vk* f ex*. (iii) The maps A ++ vx and 2 - vX are strongly increasing from their domains to C,‘(B).

LEMMA

Proof. (i) For all A, & 5 J, < Lo, 0’ is a supersolution of (2.7),. If A, < A,, then v, is a positive subsolution of (2.7), for all 1 L J., and so it follows from lemma 2.4 that there exist maximal and minimal solutions of (2.7), in the order interval [u, , tl’] for A E [A,, A’]. Since any solution v of (2.7), must satisfy u I 0’ and the above maximal solution, which we denote by ux, can be obtained from an iteration scheme starting from 0’ we have that u I v’, i.e. vx is the maximal positive solution. If A, = A,, then Bx is a positive subsolution of (2.7), for all ,J > A,. It follows as above that (2.7), has a maximal positive solution vx lying in the order interval [e,, ex] for all A E (A,, A’). (ii) Equation (2.7), has a positive subsolution 8, for all A > A,. If A* > lo, then v* # Ox* is a supersolution of (2.7), for all A I i* and, if A* = A’, 0’ is a supersolution. Hence, we get a minimal positive solution ux of (2.7), as in (i) above. If I* > lo, since u(u*) # 0, we must have k,(cu*) < a and, hence, A,(cu,) 5 hi(cv*) < a which implies ux # 8’. In the case A* = A’, for A < A*, since vx 5 Ox, A,(cvA) 5 AI < A,(cOxo) = a and so ux # 8’.

Reaction-diffusion

7

systems

(iii) The fact that 1 - ux and J. c uh are strong increasing solution arguments and the strong maximum principle.

follows

from

sub- and super-

The above lemma shows that a maximal positive solution ux is defined for (2.7), for all A E (13,) A’) (or [A,, A’) if A, < A,). For 1 2 Jo we define Y’ = 8’ so that ux is the maximal positive solution for all ,4 > A*. Similarly we extend the definition of z+,by defining ux = 0 for A IL,. 3. CONTINUA

OF SOLUTIONS

A straightforward local bifurcation analysis shows that bifurcation occurs from the branch theory and maximum of zero solutions of (2.7), when A = Ao. Using global bifurcation principle arguments as in [l] it can be shown that (2.7), has in R x C~@) a closed connected set of nonnegative solutions S+ which joins (&,, 0) to co in the (A, v) plane such that S+ fl R x (O] = ((A,, 0)). infinity. Thus, it follows It can be proved as in [l] that A + 00 as (A, u) E S+ approaches from lemma 2.3 (iv) that S+ can approach infinity only by joining up with the continuum U = ((A, 0’): ,Y > A,). Clearly we have that S+ fl U = ((A, 0’): J. 2 A’]. We define s,+ = s+\((n, It is easy to see that if (A, v) E S,f\((A,, (2.7), . We now obtain more information require the following simple general

BX): /I > 201.

0), (no, 0’“)) then u is a positive

about the structure of Si. result about connectedness.

nontrivial

In order

solution

of

to do so we shall

LEMMA3.1. Suppose that X is a Banach space, C is a connected set in X and &2is an open set in X such that %2 II C consists of a single point. Then C\a is a connected set. Proof, that

Suppose

that C\0 c\fi

is not connected. = c, u c,,

Then we can find nonempty C, n c, = c,

n C, = 0.

Suppose (x0) = aa n C. Then x0 E C\Q and, hence, x0 belongs and C,, say C, . Denote C3 = (Q U C,) fl C. Then C3

set C, and C, such

to one and only one of C,

n c, E @ n c,) u (C, n c,) = d n c, = (t2 n c,) u (82 n c,) = ai2 n c, = (22 n c) n c, = (x0)n c, = 0

c,nc,~(~nnC,)u(c,nC,)=~nc,. Since Q is open, Q fl C, = 0 implies Q n Cl = 0 and, hence, C, fl CZ = (2. Since C = C, U C3, the above discussion shows that C is not connected which contradicts our assumption. We shall also require

the following

technical

result.

8

Y.Du and K. J. BROWN

LEMMA3.2. Suppose that u is the maximal positive solution of (2.7), and consider the closed set A = [A,oo) x [v,oo) in R x C,‘(o). Then aA fl S+ E ((A, v)]. Proof.

Suppose (p, w) E f3A n SC.Clearly

ah = (Alx [~,q u [+J) x ab,q where a[u, 00) denotes the boundary of [u, 00) in C,‘(D). Suppose p > A. Then w E a[u, 00) and so w L u but w $4 u. Let 4 = w - u. Choose A4 > 0 such that f: v/ - At+v- ty2 + n(y)t+v + My/ is increasing for v I I+Y I w. Then -Ad

+ (gf?” + M)$I = ,uw - w2 + n(w)w + Mw - (Au - v* + n(v)u + Mu) >f(w)

-f(v)

2 0.

This together with 41ao = 0 imply by the strong maximum principle that 4 % 0, i.e. w 9 u. Thus, w $ d[u, co). Hence, we must have that p = 1. Since u is the maximal positive solution of (2.7),, it follows that w I v and so w = v. Thus, the proof is complete. Similarly it can be proved that if v is the minimal positive solution of (2.7), and A = (-co, A] x (-00, v] then aA n S+ G ((A, v)]. THEOREM3.1. $

is connected.

Proof. Let A = [A’, co) x [OX”,03). Clearly ((A, 8’): 2 > no) E int A and by lemma 2.3 if v is a solution of (2.7)$ with v # 0’ then v $ oh’, i.e. (A, v) g int A. Hence, Si = S+\int A. Clearly ((A*, Bx )) E S+ II aA. By lemma 3.2 S+ fl aA E ((A’, &‘“)). Hence, Sf rl aA = ((A’, @‘)l. Thus, it follows from lemma 3.1 that S+\int A is connected, i.e. S: is connected.

Thus, S,’ is a continuum in R x Cd(D) containing the points (A,, 0) and (Lo, 0’“). At all other points (A, v) in Sof v is a positive solution of (2.7), such that U(V) f 0, i.e. (u(v), v) is a positive solution of the system (2.3), . The solutions (A,, 0) and (A’, 0’“) of (2.7),0 and (2.7),0 correspond to semitrivial positive solutions of (2.3), and S$ corresponds to the continuum of solutions joining the two branches of semitrivial solutions discussed in [l]. It follows as in [l] from simple considerations of connectedness that (2.7), has a positive solution (corresponding to a point on S,+ ) for all 1 lying between A0 and 1’. Global bifurcation alone can tell us little more about S:. If, however, we also make use of the results obtained from monotonicity arguments in the previous section we can obtain much more information about SC, e.g. S,+ extends across the entire A range from Iz, to A” and contains all of the maximal and minimal solutions of lemma 2.6. We shall require the following closed sets in R x C,‘(D) Ah = (-m,n]

x (-03, u,];

THEOREM3.2. (i) (A, vx) E S,C for A, < A < i*. (ii) (A, v’) E S,+ for A* < A < i1’.

Ax = [1,~)

x [u’,m).

Reaction-diffusion

systems (b)

I

--*

w

Fig. 2.

Proof. Suppose I, < I < A*. Clearly (&,, 0) E $ Il int A,. Also (A’, @“) E S: it follows (A’, 0’“) $ int Ax, as by lemma 2.3 (v) uk $k I!?“. Since Sz is connected, S: Il aAk # 0. Hence, by lemma 3.2 S: fl aAx = {(A, v,)], i.e. (A, vx) E S:. The proof of (ii) is similar.

but that

Bifurcation diagrams consistent with the above theorem are shown in Fig. 2 in which the maximal and minimal solutions are shown by solid lines and other nontrivial solutions by dashed lines. These diagrams suggest in turn other theorems about the existence of solutions. However, such theorems cannot be proved by simply considering bifurcation diagrams such as these shown, as it is possible that S; has a more complicated form than the smooth curves in the diagrams. The next theorem proves the existence of “small” solutions of (2.7),. THEOREM 3.3. Suppose 1, < L < I,. Then there exists a positive u # vx (recall that we have defined ux = 0’ for A I 1’).

solution

u of (2.7), such that

Proof. We consider first the case where A, < Lo. By lemma 2.6, (2.7), has a maximal positive solution ux % ux*. Using lemma 3.2 we easily see that SL fl aA’* = ](A,, d*)) and so by lemma 3.1 $\int Ax* is connected. Since (A,, 0) and (A,, d*) are contained in Si\int Ax* it follows that there exists u such that (J., u) E S,f\int Ax*. Thus, u # Y’. A simpler argument suffices when A.+ = A’; in this case ux = 8’ and if (A, u) lies on the continuum S: which joins (Ao, 0) and (A,, OX*) then we must have that u # u’. A similar result holds for ,l” < 1 < A*. The next theorem deals with the existence of positive solutions when A = 13,. The bifurcation diagrams in Fig. 2 show that in general positive solutions may or may not exist in this case. THEOREM 3.4. Suppose

A* < A., < 1*. Then (2.7),0 has a nontrivial

positive

solution

u.

Proof. We must consider various possibilities for the position of A’. If Lo > Lo, then by theorem 3.2 equation (2.7), has a positive solution uh for all J E (A,, 1’) and so (2.7),0 has a positive solution.

Y.

10

Du and K. J.BROWN

Suppose now that A0 < A,. Then from theorem 3.3 there is a positive solution of (2.7), for all J E (A’, A*) and so (2.7& has a positive solution. Finally we consider the case where A0 = A,. We may assume that limx,x, ux = 0 and lim X_hOUX = 0’” as otherwise compactness arguments could be used to prove the existence of an appropriate solution when A = Lo. Choose E > 0 so that Y”-~ 2 u~,+~. It follows from lemma 3.1 as before that Sc\int AXO+Eis connected. Since (S,f\int Ax,+E) II aAX’-’ = set by lemma 3.1. Moreover, ((1’ - E, vxo-‘)I, ($\int AhO+,)\int Ax’-’ is also a connected it contains the points (A0 - E, uxo-‘) and (A0 + E, uX,,+J, and ($\int AhO+,)\into AX’-’ and the proof is complete.

so there

A similar result holds for the case where A* < A0 < A*. We can now give our main theorem on the existence and multiplicity 3.5. (i) There exists a nontrivial (ii) There exist at least two nontrivial min(1,, no) or max(A,, no) < A < A*.

THEOREM

positive positive

exists

of solutions

(A,, u) E

of (2.7), .

solution u of (2.7), for all A, A* < A < A*. solutions of (2.7), when either A, < A <

Proof. Lemma 2.6 shows that (2.7), has a positive solution for ,J E (A,, Lo) and J. E (A,, A*). Since S: joins (A,, 0) and (A’, @‘“), (2.7), has a positive solution for all A between A, and i1’. Finally the cases 1 = A0 or A = ho follow from theorem 3.4. (ii) Suppose A, < ,J < min(A,, A’]. By lemma 2.6 equation (2.7), has a maximal positive solution z?’ and by theorem 3.3 the equation has another solution u # u’. A similar proof can be given for the case maxlAo, no) < A < J.*. In the case where A0 < A0 more can be said about the multiplicity of solutions of (2.7), for A E (Lo, A’). The following theorem allows the possibility of a bifurcation diagram such as that shown in Fig. 3(a) but rules out the possibility of the bifurcation diagram shown in Fig. 3(b). THEOREM 3.6. Suppose A0 < A’. Then for each ,D E (Ao, A’), one of the following must occur: (i) (2.7), has a unique nontrivial positive solution for A = ,D; (ii) (2.7), has a continuum of nontrivial positive solutions for J. = ,D;

(b)

(a)

1 Y

rJ

Fig.3

three cases

Reaction-diffusion

systems

11

(iii) there exists an interval (pi, pJ C (A,,A') with pu, 5 p I ,uz and pi < pz such that for every I E (pi, ,uJ, (2.7), has at least three nontrivial positive solutions. Proof. Suppose fi E (Ao, Lo) and (2.7), has more than one positive solution for A = P. We are going to prove that case (ii) or (iii) occurs. As (2.7), has more than one positive solution for 1 = p, we must have u” I v,, d’ # u, which implies int AC n aAP = 0. Hence, (S,f\int

AJ fl aAa = (S,’ fl aA”)\(int

From lemma 3.2 we know that C = Sl\int single point set. So by lemma 3.1 3 = C\Q

A& n aAp) = S: II aAP = ((P, v’)). Ap is connected.

= s,‘\(int

If fi = int A”, then C n afi is a

A,, U int A@)

is connected. If 2 G (,~u)x C,‘(o), clearly case (ii) occurs. Otherwise there must exist ,LL, such that (Pi, vi) E g and either pi < ,D or ,D, > p. Suppose pi < ,u. Now since v”’ z vi and (pi, ui) E 3 we find vi $ v, and, hence, vW1# v,. This implies (pi, Y”‘) $ int A,, and (p, up) $ int AP1. Thus, (p,, up’), (p, vr) both belong to S,f\(int A, U int API). As tP # v, we have that int A,, n aAP1 = 0. It follows as in the proof of the connectedness of $ that S,f\(int A, U int A”‘) is connected. Since it connects (,~i, Y”‘) and (,M, uf), for each A E (,D, , ,Y) we can find (A, v) E Si\(int AP U int A@‘). Hence, (A, v,) E int A,, , (A, v ) E int A”’ and (A, u) $ int A, U int ApI are three positive solutions of (2.7),. Similarly in the case where pi > ,Y we can find three positive solutions of (2.7), for each A E MYPi). It was proved in [4] that for almost all (a, e) E R2 with a > Ai, e > A, one can find (c, g) such that (2.3), has more than one positive solution. Our final theorem shows how our earlier results also indicate that multiple solutions occur for many choices of parameters. THEOREM 3.7.For any (a, c) E R2 with a > A, and c > 0, we can find e > A,, and g > 0 such that (2.3), has more than one positive solution. Proof. Suppose a > A, and c > 0 are given. Then we can find unique e > Ai and g > 0 such that a = AI and ai = e. Now consider equation (2.7), for this choice of a, c and g. We have that Jo = Lo = e. If A* = e = A*, then (2.7), has a continuum of positive solutions for L = e, i.e. (2.7), has infinitely many positive solutions. If A* < e, then by theorem 3.5, (2.7), has at least two positive solutions for J., < I < e and similarly if e < A*, (2.7), has at least two positive solutions for e < L < A*. It is possible to find parameter values such that each of the possibilities previous theorems occur. It is shown in [3] that the system -Au

= au - u2 - uv,

XED

-Au

= Au - v2 - uu,

XED

u=v=o, has A* = A, = 1’ = A*.

XEaD

described

in the

12

Y. Du and K. J. BROWN

On the other hand by considering the direction such that A, < &, = A0 or A, = A0 < A*. Let

s = ((A,

of bifurcation

u(u), u): (A, u)

E

at A0 we can find parameters

S,‘].

It was proved in [2] by using local bifurcation theory that if (Ao, u, u) E S and (A, U, u) is close to but not equal to (no, 8”, 0) then A < A0 (respectively, A > Ao) provided (3.1) (respectively, (3.2)) holds

.r s ly3dx - gc

D

iy3dx - gc .i D where w and 6 are determined -A+

(v2q5dx< 0

(3.1)

y&l dx > 0

(3 -2)

D

i

D

by + (20” - a)4 = Bay/,

-Aw

i

XED

4 = 0,

XEaD

+ g@y/ = h(ge%,

XED

w = 0,

x E aD

ly2dx = 1,

IJ 2 0.

D

For any t, s with s, t > A, there exists a unique (Y = cr(t, s) such that t = A,(d). Since A0 = Ai and a = A,(c@‘“), it follows that g = ol(Ao, a) and c = a(a, A’). Thus, for given a we may arrange to have A0 = A0 = s by choosing g = a@, a) and c = cx(a,s). From results about the functionfi(c) in Section 2 of [4] we can see that for any to > Al, there exists MO > 0 such that (3.2) holds for (a, c, g) = (to, a(t,, s), a@, to)) provided s > MO. Thus, for such parameter values we have 1, = 1’ < A*. Similarly by using results about the functionf,(e) in Section 2 of [4] it can be proved that for any so > A, there exists No > 0 such that (3.1) holds for (a, c, g) = (t, c.u(t,so), cx(so, t)) provided t > No and so in this case we have A* < A, = Lo. Acknowledgements-The authors are grateful to Professor E. N. Dancer for his valuable comments on the original draft of this paper. This work was done while the first author was on leave from Shandong University, People’s Republic of China. He would like to thank the Mathematics Department of Heriot-Watt University for their kind hospitality. The support from the Sino-British Friendship Scholarship is also gratefully acknowledged.

REFERENCES BLAT J. & BROWN K. J., Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. R. Sot. Edinb. 97A, 21-34 (1984). CANTRELL R. & COSNER C., On the steady-state problem for the Volterra-Lotka competition model with diffusion, Houston J. math. 13, 337-352 (1987). COSNER C. & LAZER A. C., Stable coexistence state in the Voherra-Lotka competition model with diffusion, SIAM J. math. Analysis 44, 1112-1132 (1984). Dancer E. N., On the existence and uniqueness of positive solutions for competing species models with diffusion, Trans. Am. math. Sot. 326, 829-859 (1991).

Reaction-diffusion

systems

13

5. DANCER E. N., On positive solutions of some pairs of differential equations II, J. &‘jJ Eqns 60, 236-258 (1985). 6. KORMAN P. & LEUNG A., On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion, Applicable Analysis 26, 145-160 (1987). 7. SATTINGER D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. math. J. 21, 979-1000 (1972).